When \(k = n-1\) , equation (REF ) is just the smooth case of Orlicz-Minkowski problem. The Orlicz-Minkowski problem is a basic problem in the Orlicz-Brunn-Minkowski theory in
convex geometry. It is a generalization of the classical Minkowski problem which
asks what are the necessary and sufficient conditions for
a Borel measure on the unit sphere \(\mathbb {S}^{n-1}\)  to be a multiple of the Orlicz surface area
measure of a convex body in \(\mathbb {R}^n\) . In [1]}, Haberl,
Lutwak, Yang & Zhang studied the even case of the Orlicz-Minkowski problem. After that,
the Orlicz-Minkowski problem attracted great attention from many scholars,
see for example [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
